1. Technical Field
The invention relates to a method of reducing the calculation time for a numerical calculation for a computer implemented superposition model.
2. Discussion of the Prior Art
It is known to compute a superposition model for a waveform function G(txe2x88x92xcfx84i), as shown in Formula 1. A sampled arrangement G(I) is interpolated by shifting by xcfx84i for the respective suffix i to compute an arrangement relative to the function G(txe2x88x92xcfx84i) throughout the domain of the variable t.                               Y          ⁢                      (            t            )                          =                              ∑                          i              =              1                                      N              i                                ⁢                                    a              i                        ·                          G              ⁢                              (                                  t                  -                                      τ                    i                                                  )                                                                        FORMULA        ⁢                  xe2x80x83                ⁢        I            
where:
Y(t): Superposed waveform where (t) time represents a variable, for example, seismic waves, sound waves, light waves, electromagnetic waves, temperature, voltage, electric current, fluid pressure; additionally, position, displacement, temperature, voltage, electric flow, fluid pressure, or other physical quantities can be substituted for the time.
G(t): Waveform where (t) time represents a variable, for example, seismic waves, sound waves, light waves, electromagnetic waves, temperature, voltage, electric current, fluid pressure; additionally, record of position, displacement, temperature, voltage, electric flow, fluid pressure, or other physical quantities can be substituted for the time.
a: Coefficient
i: Suffix
xcfx84: Time lag
t: Time
Ni: Number of superposition
{dot over ( )}: Product
The above-described method has some shortcomings. For example:
The function G(t) is a waveform (see FIG. 1) comprising the time t as a variable. The function G(t), however, must first be digitized to be processed by a digital computer. That is, data are processed in the computer in accordance with the arrangement G(I) sampled at xcex94t, not the function G(t). Accordingly, the function G(txe2x88x92xcfx84i) must be determined by interpolation, e.g. by shifting the arrangement G(I) by xcfx84i. This interpolation requires significant processing time because the respective time t, i.e. all arrangements, must be interpolated.
Further, the function G(txe2x88x92xcfx84i) with regard to the respective suffix i, must be multiplied by a coefficient ai. For the purpose of simplifying the discussion herein, the arrangement of the function G(txe2x88x92xcfx84i) comprising xcfx84i is interpolated by shifting the arrangement G(I) of the function G(t) for xcfx84i. All arrangements hereinafter are shown in this manner. The function G(txe2x88x92xcfx84i) is treated as a function in a mathematical formula, while it is also treated as an arrangement in a calculation. To simplify the discussion herein, arrangements are also described as functions in some cases. As discussed above, significant processing time is required to determine the function arrangement for the respective suffix i. Therefore, this type of calculation is not practically implemented in a personal computer.
A superposition model may be used, for example, for the prediction of large seismic waves from data describing small seismic waves. FIG. 2 illustrates a method of predicting an earthquake at the observation points OP. When the waveform G(t) of a small dislocation earthquake SE is observed, displacement occurred on the starting point SP are transferred in different directions with a time lag xcfx84, finally reaching an arrival point AP. This results in a large earthquake LE in which displacement is amplified over a vast range. See Eiji Kojima, A Semi-Empirical Method for Synthesizing Intermediate-Period Strong Ground Motions, Doctoral Thesis, Tohoku University (August 1996).
In the foregoing prediction model, the delay is xcfx84lm, which is the traveling time of the seismic waves from the small area (l, m) to the observation point OP, provided that the displacement area is divided in small pieces and that the seismic waves occur at a respective small area (l, m) with a defined delay.
Formula 2, may be used to predict the large seismic wave Y(t), taking into consideration the distance decrement, radioactive characteristics, and slip conditions at the respective area (l, m). A waveform of a small seismic wave G(t) may therefore be processed for superposition to predict a large seismic wave Y(t).                               Y          ⁢                      (            t            )                          =                              ∑                          l              =              1                                      N              1                                ⁢                      xe2x80x83                    ⁢                                    ∑                              m                =                1                                            N                m                                      ⁢                          xe2x80x83                        ⁢                                          ∑                                  k                  =                  1                                                  N                  k                                            ⁢                              xe2x80x83                            ⁢                                                (                                                            X                      lomo                                        /                                          X                                              l                        ⁢                                                  xe2x80x83                                                ⁢                        m                                                                              )                                ·                                  (                                                            R                                              l                        ⁢                                                  xe2x80x83                                                ⁢                        m                                                              /                                          R                      lomo                                                        )                                ·                                  G                  ⁢                                      (                                          t                      -                                              τ                                                  l                          ⁢                                                      xe2x80x83                                                    ⁢                          m                                                                    -                                                                        (                                                      k                            -                            1                                                    )                                                ·                        ψ                                                              )                                                                                                          FORMULA        ⁢                  xe2x80x83                ⁢        2            
Where:
Y(t): Large seismic wave representing a ground motion in a large earthquake.
G(t): Small seismic wave representing a ground motion in a small earthquake.
Xlm: Distance from the observation point to the small area created by dividing the displacement of a large earthquake.
Xlomo: Distance from the seismic center of the small earthquake.
Rlm: Radioactive characteristic to the observation point of the small area created by dividing the displacement of a large earthquake.
Rlomo: Radioactive characteristic to the observation point of a small earthquake.
xcfx84: Time lag
t: Time
l, m, k: Suffix
Nl, Nm, Nk: Number of superposition "psgr": Time lag of slippage in a small earthquake.
This invention resolves the above disadvantages of the prior art techniques by providing a method for reducing the calculation time for the numerical calculation of a computer implemented superposition model.
In a first embodiment of the invention, a method is provided for reducing the calculation time of a numerical calculation for a computer implemented superposition model. This method includes a step for calculating a model of superposing the function with shifted values for a variable using a computer equipped with an input unit, an output unit, and a memory unit. A model operator is formed by superposing the delta function in the same manner as the superposed function model to determine the composition product of the model operator and the function.
In a second embodiment of the invention, a method is provided for reducing the calculation time of a numerical calculation for a computer implemented superposition model, in which the model is computed plural times.
In a third embodiment of this invention, a method is provided for reducing the calculation time for a numerical calculation for a computer implemented superposition function model, in which a variable of the function is time, location, temperature, or other physical quantities.
In a fourth embodiment of this invention, a medium is provided for use by computer and on which a program is stored for reducing the calculation time of a computer for the numerical calculation of a superposition model. Using the program stored on the medium, a model of superposing the function having shifted values for a variable is calculated by a computer equipped with an input unit, an output unit, and a memory unit. A model operator is formed by superposing a delta function in the same manner as the superposed function model to determine the composition product of the model operator and the function.
In a fifth embodiment of the invention, a medium is provided for use by a computer and having therein a program for reducing the calculation time of a numerical calculation of a superposition, the model is computed plural times.
In a sixth embodiment of the invention, a medium is provided for storing a program for reducing the calculation time of a numerical calculation of a superposition model, a variable of the function is time, location, temperature, or other physical quantities.